Yvon Lafranche

Fast semi-analytic computation of elastic edge singularities

Abstract The singularities that we consider are the characteristic non-smooth solutions of the equations of linear elasticity in piecewise homogeneous media near two-dimensional corners or three-dimensional edges. We describe here a method to compute their singularity exponents and the associated angular singular functions. We present the implementation of this method in a program whose input data are geometrical data, the elasticity coefficients of each material involved and the type of boundary conditions (Dirichlet, Neumann or mixed conditions). Our method is particularly useful with anisotropic materials and allows to “follow” the dependency of singularity exponents along a curved edge.

Combining analytic preconditioner and Fast Multipole Method for the 3-D Helmholtz equation

The paper presents a detailed numerical study of an iterative solution to 3-D sound-hard acoustic scattering problems at high frequency considering the Combined Field Integral Equation (CFIE). We propose a combination of an OSRC preconditioning technique and a Fast Multipole Method which leads to a fast and efficient algorithm independently of both a frequency increase and a mesh refinement. The OSRC-preconditioned CFIE exhibits very interesting spectral properties even for trapping domains. Moreover, this analytic preconditioner shows highly-desirable advantages: sparse structure, ease of implementation and low additional computational cost. We first investigate the numerical behavior of the eigenvalues of the related integral operators, CFIE and OSRC-preconditioned CFIE, in order to illustrate the influence of the proposed preconditioner. We then apply the resolution algorithm to various and significant test-cases using a GMRES solver. The OSRC-preconditioning technique is combined to a Fast Multipole Method in order to deal with high-frequency 3-D cases. This variety of tests validates the effectiveness of the method and fully justifies the interest of such a combination.

A knot removal strategy for scattered data in R 2

We present a strategy for reducing the number of nodes for the representation of a piecewise polynomial approximation of a function defined on scattered data, without perturbing the approximation more than a given tolerance. The method removes some (or all) of the interior knots. The number and the location of these knots are determined automatically.

Dirichlet Spectrum of the Fichera Layer

We investigate the spectrum of the three-dimensional Dirichlet Laplacian in a prototypal infinite polyhedral layer, that is formed by three perpendicular quarter-plane walls of constant width joining each other. Alternatively, this domain can be viewed as an octant from which another “parallel” octant is removed. It contains six edges (three convex and three non-convex) and two corners (one convex and one non-convex). It is a canonical example of non-smooth conical layer. We name it after Fichera because near its non-convex corner, it coincides with the famous Fichera cube that illustrates the interaction between edge and corner singularities. This domain could also be called an octant layer. We show that the essential spectrum of the Laplacian on such a domain is a half-line and we characterize its minimum as the first eigenvalue of the two-dimensional Laplacian on a broken guide. By a Born–Oppenheimer type strategy, we also prove that its discrete spectrum is finite and that a lower bound is given by the ground state of a special Sturm–Liouville operator. By finite element computations taking singularities into account, we exhibit exactly one eigenvalue under the essential spectrum threshold leaving a relative gap of 3%. We extend these results to a variant of the Fichera layer with rounded edges (for which we find a very small relative gap of 0.5%), and to a three-dimensional cross where the three walls are full thickened planes.

Node Insertion and Node Deletion for Radial Basis Functions

The present paper is a contribution to data reduction in the multivariate case. In this direction, there are quite a few methods like ours dealing with scattered data points. Besides piecewise polynomial splines, radial basis functions (RBF splines) provide another natural generalization of univariate splines. Considering a set A of distinct scattered data points and a given tolerance c, our aim is to extract a subset A ⊂ A such that the RBF spline σ A stays within a tolerance A from the RBF spline σ A built upon the entire set A of data points. We show that for the usual RBF splines, data reduction can be considered from two different points of view: a priori reduction which amounts to adding nodes one after the other, starting from a small subset of A, thus refering to a rough approximation of σ A , and a posteriori reduction, which in contrast consists of deleting nodes one after the other from the ultimate spline σ A

KNOT REMOVAL FOR SCATTERED DATA

We present a strategy for reducing the number of knots for the representation of a piecewise polynomial approximation of a function defined on scattered data, without perturbing the approximation more than a given tolerance. The method removes some (or all) of the interior knots. The number and location of these knots are determined automatically. Applications are in approximation of data, data storage, and image reconstruction.